Solve Maxwell's equations (Gauss's law for electric fields, Gauss's law for magnetic fields, Faraday's law of electromagnetic induction, and Ampère's law with Maxwell's addition) in a non-Cartesian coordinate system, such as cylindrical or spherical coordinates. Find the electric and magnetic field distributions in this system.
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To solve Maxwell's equations in a non-Cartesian coordinate system, let's consider the equations in cylindrical coordinates as an example.
In cylindrical coordinates, the radial distance from the origin is denoted by r, the azimuthal angle by θ, and the axial distance by z.
The first equation is Gauss's law for electric fields:
∇·E = ρ/ε₀
In cylindrical coordinates, this equation becomes:
(1/r) ∂(rEᵣ) / ∂r + (1/r) ∂Eₜ / ∂θ + ∂Ez / ∂z = ρ/ε₀
The second equation is Gauss's law for magnetic fields:
∇·B = 0
In cylindrical coordinates, this equation becomes:
(1/r) ∂(rBᵣ) / ∂r + (1/r) ∂Bₜ / ∂θ + ∂Bz / ∂z = 0
The third equation is Faraday's law of electromagnetic induction:
∇×E = -∂B / ∂t
In cylindrical coordinates, this equation becomes:
(1/r) ∂(rEz) / ∂θ - ∂Er / ∂z = -(∂Bz / ∂t)
The fourth equation is Ampère's law with Maxwell's addition:
∇×B = μ₀J + ε₀μ₀ ∂E / ∂t
In cylindrical coordinates, this equation becomes:
(1/r) ∂(rBz) / ∂θ - ∂Br / ∂z = μ₀Jz + ε₀μ₀ (∂Ez / ∂t)
These equations can be solved using appropriate boundary conditions and the known sources/charges in the system. The solutions will provide the electric and magnetic field distributions in the cylindrical coordinate system.
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